 3.3.4.1: solve the system Ax b using the given LUfactorization of A.
 3.3.5.1: let S be the collection of vectors in 2 that satisfy the given prop...
 3.3.1.1: compute the indicated matrices (ifpossible). A 2D
 3.3.6.1: Let TA : 2 S 2 be the matrix transformation correspondingto Find TA...
 3.3.2.1: X 2A 3B O
 3.3.7.1: let be the transition matrixfor a Markov chain with two states. Let...
 3.3.3.1: find the inverse of the given matrix (if it exists) using Theorem 3.8.
 3.3.4.2: solve the system Ax b using the given LUfactorization of A.
 3.3.5.2: let S be the collection of vectors in 2 that satisfy the given prop...
 3.3.1.2: compute the indicated matrices (ifpossible). 2D 5A
 3.3.6.2: Let TA : 2 S 3 be the matrix transformation correspondingto . Find ...
 3.3.2.2: 3X A 2B
 3.3.7.2: let be the transition matrixfor a Markov chain with two states. Let...
 3.3.3.2: find the inverse of the given matrix (if it exists) using Theorem 3.8.
 3.3.4.3: solve the system Ax b using the given LUfactorization of A.
 3.3.5.3: let S be the collection of vectors in 2 that satisfy the given prop...
 3.3.1.3: compute the indicated matrices (ifpossible). B C
 3.3.6.3: T c xyd cx yx yd
 3.3.2.3: 21A 2B2 3X
 3.3.7.3: let be the transition matrixfor a Markov chain with two states. Let...
 3.3.3.3: find the inverse of the given matrix (if it exists) using Theorem 3.8.
 3.3.4.4: solve the system Ax b using the given LUfactorization of A.
 3.3.5.4: let S be the collection of vectors in 2 that satisfy the given prop...
 3.3.1.4: compute the indicated matrices (ifpossible). . B2
 3.3.6.4: T cxyd x 2yx3x 7yT c x
 3.3.2.4: 21A B 2X2 31X B22
 3.3.7.4: let be the transition matrixfor a Markov chain with two states. Let...
 3.3.3.4: find the inverse of the given matrix (if it exists) using Theorem 3.8.
 3.3.4.5: solve the system Ax b using the given LUfactorization of A.
 3.3.5.5: let S be the collection of vectors in 3 that satisfy the given prop...
 3.3.1.5: compute the indicated matrices (ifpossible). D BC
 3.3.6.5: T x y z xx yx y z
 3.3.2.5: write B as a linear combination of the othermatrices, if possible.
 3.3.7.5: let be the transition matrixfor a Markov chain with three states. L...
 3.3.3.5: find the inverse of the given matrix (if it exists) using Theorem 3.8.
 3.3.4.6: solve the system Ax b using the given LUfactorization of A.
 3.3.5.6: let S be the collection of vectors in 3 that satisfy the given prop...
 3.3.1.6: compute the indicated matrices (ifpossible). BC B
 3.3.6.6: give a counterexample to show that thegiven transformation is not a...
 3.3.2.6: write B as a linear combination of the othermatrices, if possible.
 3.3.7.6: let be the transition matrixfor a Markov chain with three states. L...
 3.3.3.6: find the inverse of the given matrix (if it exists) using Theorem 3.8.
 3.3.4.7: find an LU factorization of the given matrix.
 3.3.5.7: let S be the collection of vectors in 3 that satisfy the given prop...
 3.3.1.7: compute the indicated matrices (ifpossible).E 1AF2
 3.3.6.7: give a counterexample to show that thegiven transformation is not a...
 3.3.2.7: write B as a linear combination of the othermatrices, if possible.
 3.3.7.7: let be the transition matrixfor a Markov chain with three states. L...
 3.3.3.7: find the inverse of the given matrix (if it exists) using Theorem 3.8.
 3.3.4.8: find an LU factorization of the given matrix.
 3.3.5.8: let S be the collection of vectors in 3 that satisfy the given prop...
 3.3.1.8: compute the indicated matrices (ifpossible). E 1AF2
 3.3.6.8: give a counterexample to show that thegiven transformation is not a...
 3.3.2.8: write B as a linear combination of the othermatrices, if possible.
 3.3.7.8: Suppose that the weather in a particular region behaves according t...
 3.3.3.8: find the inverse of the given matrix (if it exists) using Theorem 3.8.
 3.3.4.9: find an LU factorization of the given matrix.
 3.3.5.9: Prove that every line through the origin in 3 is a subspace of 3 .
 3.3.1.9: compute the indicated matrices (ifpossible). E 1AF2 F
 3.3.6.9: give a counterexample to show that thegiven transformation is not a...
 3.3.2.9: find the general form of the span of the indicated matrices, as in ...
 3.3.7.9: Data have been accumulated on the heights ofchildren relative to th...
 3.3.3.9: find the inverse of the given matrix (if it exists) using Theorem 3.8.
 3.3.4.10: find an LU factorization of the given matrix.
 3.3.5.10: Suppose S consists of all points in 2 that are on the xaxis or the...
 3.3.1.10: compute the indicated matrices (ifpossible).F 1AF2
 3.3.6.10: give a counterexample to show that thegiven transformation is not a...
 3.3.2.10: find the general form of the span of the indicated matrices, as in ...
 3.3.7.10: Data have been accumulated on the heights of children relative to t...
 3.3.3.10: find the inverse of the given matrix (if it exists) using Theorem 3.8.
 3.3.4.11: find an LU factorization of the given matrix.
 3.3.5.11: determine whether b is in col(A) and whether w is in row(A), as in ...
 3.3.1.11: compute the indicated matrices (ifpossible). FE
 3.3.6.11: find the standard matrix of the lineartransformation in the given e...
 3.3.2.11: find the general form of the span of the indicated matrices, as in ...
 3.3.7.11: A study of pion (pine) nut crops in the American southwest from 194...
 3.3.3.11: solve the given system using themethod of Example 3.25.
 3.3.4.12: find an LU factorization of the given matrix.
 3.3.5.12: determine whether b is in col(A) and whether w is in row(A), as in ...
 3.3.1.12: compute the indicated matrices (ifpossible). EF
 3.3.6.12: find the standard matrix of the lineartransformation in the given e...
 3.3.2.12: find the general form of the span of the indicated matrices, as in ...
 3.3.7.12: Robots have been programmed to traverse the maze shown in Figure 3....
 3.3.3.12: solve the given system using themethod of Example 3.25.
 3.3.3.13: Let and(a) Find A1 and use it to solve the three systemsAx b1, Ax b...
 3.3.4.13: Generalize the definition of LU factorization to nonsquarematrices ...
 3.3.5.13: In Exercise 11, determine whether w is in row(A), using the method ...
 3.3.1.13: compute the indicated matrices (ifpossible). B DA AD TCT 1CB2TFE
 3.3.6.13: find the standard matrix of the lineartransformation in the given e...
 3.3.2.13: determine whether the given matrices are linearly independent.
 3.3.7.13: Let j denote a row vector consisting entirely of 1s. Provethat a no...
 3.3.3.14: Prove Theorem 3.9(b).
 3.3.4.14: Generalize the definition of LU factorization to nonsquarematrices ...
 3.3.5.14: In Exercise 12, determine whether w is in row(A) using the method d...
 3.3.1.14: compute the indicated matrices (ifpossible). DA AD T
 3.3.6.14: find the standard matrix of the lineartransformation in the given e...
 3.3.2.14: determine whether the given matrices are linearly independent.
 3.3.7.14: (a) Show that the product of two 2 2 stochasticmatrices is also a s...
 3.3.3.15: Prove Theorem 3.9(d).
 3.3.4.15: For an invertible matrix with an LU factorization A LU,both L and U...
 3.3.5.15: If Ais the matrix in Exercise 11, is in null(A)?
 3.3.1.15: compute the indicated matrices (ifpossible).A3
 3.3.6.15: how that the given transformationfrom 2 to 2 is linear by showing t...
 3.3.2.15: determine whether the given matrices are linearly independent.
 3.3.7.15: In Exercise 9, if Monday is a dry day, what is the expected number ...
 3.3.3.16: Prove that the n n identity matrix In is invertible and that In.
 3.3.4.16: For an invertible matrix with an LU factorization A LU,both L and U...
 3.3.5.16: . If Ais the matrix in Exercise 12, is in null(A)?
 3.3.1.16: compute the indicated matrices (ifpossible).1I2 A22 A
 3.3.6.16: how that the given transformationfrom 2 to 2 is linear by showing t...
 3.3.2.16: determine whether the given matrices are linearly independent.
 3.3.7.16: In Exercise 10, what is the expected number of generations until a ...
 3.3.3.17: (a) Give a counterexample to show that (AB) 1 Z A1 B1 in general. (...
 3.3.4.17: use the approach just outlined to findA1 for the given matrix. Comp...
 3.3.5.17: give bases for row(A), col(A), and null(A).
 3.3.1.17: Give an example of a nonzero 2 2 matrix A suchthat A2 O.
 3.3.6.17: how that the given transformationfrom 2 to 2 is linear by showing t...
 3.3.2.17: Prove Theorem 3.2(a)(d).
 3.3.7.17: In Exercise 11, if the pion nut crop is fair one year, what is the ...
 3.3.3.18: .By induction, prove that if A1, A2,..., An are invertible matrices...
 3.3.4.18: use the approach just outlined to findA1 for the given matrix. Comp...
 3.3.5.18: give bases for row(A), col(A), and null(A).
 3.3.1.18: Let Find 2 2 matrices B and C suchthat AB AC but B Z C.
 3.3.6.18: how that the given transformationfrom 2 to 2 is linear by showing t...
 3.3.2.18: Prove Theorem 3.2(e)(h)
 3.3.7.18: In Exercise 12, starting from each of the other junctions, what is ...
 3.3.3.19: Give a counterexample to show that (A B) 1 Z A1 B1 in general. In E...
 3.3.4.19: write the given permutation matrix as aproduct of elementary (row i...
 3.3.5.19: give bases for row(A), col(A), and null(A).
 3.3.1.19: A factory manufactures three products (doohickies,gizmos, and widge...
 3.3.6.19: The three types of elementary matrices give rise to fivetypes of 2 ...
 3.3.2.19: Prove Theorem 3.3(c).
 3.3.7.19: determine which of the matrices are exchange matrices. For those th...
 3.3.3.20: solve the given matrix equation for X.Simplify your answers as much...
 3.3.4.20: write the given permutation matrix as aproduct of elementary (row i...
 3.3.5.20: give bases for row(A), col(A), and null(A).
 3.3.1.20: Referring to Exercise 19, suppose that the unit cost ofdistributing...
 3.3.6.20: find the standard matrix of the givenlinear transformation from 2 t...
 3.3.2.20: Prove Theorem 3.3(d).
 3.3.7.20: determine which of the matrices are exchange matrices. For those th...
 3.3.3.21: solve the given matrix equation for X.Simplify your answers as much...
 3.3.4.21: write the given permutation matrix as aproduct of elementary (row i...
 3.3.5.21: find bases for row(A) and col(A) in the given exercises using AT . ...
 3.3.1.21: write the given system of linear equationsas a matrix equation of t...
 3.3.6.21: find the standard matrix of the givenlinear transformation from 2 t...
 3.3.2.21: Prove the half of Theorem 3.3(e) that was not proved in the text
 3.3.7.21: determine which of the matrices are exchange matrices. For those th...
 3.3.3.22: solve the given matrix equation for X.Simplify your answers as much...
 3.3.4.22: write the given permutation matrix as aproduct of elementary (row i...
 3.3.5.22: find bases for row(A) and col(A) in the given exercises using AT . ...
 3.3.1.22: write the given system of linear equationsas a matrix equation of t...
 3.3.6.22: find the standard matrix of the givenlinear transformation from 2 t...
 3.3.2.22: Prove that, for square matrices A and B, AB BA if and only if (A B)...
 3.3.7.22: determine which of the matrices are exchange matrices. For those th...
 3.3.3.23: solve the given matrix equation for X.Simplify your answers as much...
 3.3.4.23: find a PTLU factorization of the givenmatrix A.
 3.3.5.23: find bases for row(A) and col(A) in the given exercises using AT . ...
 3.3.1.23: Use the matrixcolumn representation of the productto write each co...
 3.3.6.23: find the standard matrix of the givenlinear transformation from 2 t...
 3.3.2.23: if find conditions on a, b,c, and d such that AB BA.
 3.3.7.23: determine which of the matrices are exchange matrices. For those th...
 3.3.3.24: In each case, find an elementary matrix E that satisfies the given ...
 3.3.4.24: find a PTLU factorization of the givenmatrix A.
 3.3.5.24: find bases for row(A) and col(A) in the given exercises using AT . ...
 3.3.1.24: Use the rowmatrix representation of the product towrite each row o...
 3.3.6.24: find the standard matrix of the givenlinear transformation from 2 t...
 3.3.2.24: if find conditions on a, b,c, and d such that AB BA.
 3.3.7.24: determine which of the matrices are exchange matrices. For those th...
 3.3.3.25: In each case, find an elementary matrix E that satisfies the given ...
 3.3.4.25: find a PTLU factorization of the givenmatrix A.
 3.3.5.25: Explain carefully why your answers to Exercises 17 and 21 are both ...
 3.3.1.25: Compute the outer product expansion of AB
 3.3.6.25: find the standard matrix of the givenlinear transformation from 2 t...
 3.3.2.25: if find conditions on a, b,c, and d such that AB BA.
 3.3.7.25: determine which of the matrices are exchange matrices. For those th...
 3.3.3.26: In each case, find an elementary matrix E that satisfies the given ...
 3.3.4.26: Prove that there are exactly n! permutationmatrices.
 3.3.5.26: Explain carefully why your answers to Exercises 18 and 22 are both ...
 3.3.1.26: Use the matrixcolumn representation of the productto write each co...
 3.3.6.26: Let / be a line through the origin in 2, P/ the lineartransformatio...
 3.3.2.26: Find conditions on a, b,c, and d such
 3.3.7.26: determine which of the matrices are exchange matrices. For those th...
 3.3.3.27: In each case, find an elementary matrix E that satisfies the given ...
 3.3.4.27: solve the system Ax b using the givenfactorization A PTLU. Because ...
 3.3.5.27: find a basis for the span of the given vectors
 3.3.1.27: Use the rowmatrix representation of the product towrite each row o...
 3.3.6.27: apply part (b) or (c) of Exercise 26 tofind the standard matrix of ...
 3.3.2.27: Find conditions on a, b, c, and d such that commutes with every 2 2...
 3.3.7.27: determine whether the given consumption matrix is productive.
 3.3.3.28: In each case, find an elementary matrix E that satisfies the given ...
 3.3.4.28: solve the system Ax b using the givenfactorization A PTLU. Because ...
 3.3.5.28: find a basis for the span of the given vectors
 3.3.1.28: Compute the outer product expansion of BA.
 3.3.6.28: apply part (b) or (c) of Exercise 26 tofind the standard matrix of ...
 3.3.2.28: Prove that if AB and BA are both defined, then AB and BA are both s...
 3.3.7.28: determine whether the given consumption matrix is productive.
 3.3.3.29: In each case, find an elementary matrix E that satisfies the given ...
 3.3.4.29: Prove that a product of unit lower triangular matricesis unit lower...
 3.3.5.29: find a basis for the span of the given vectors
 3.3.1.29: assume that the product AB makessense. Prove that if the columns of...
 3.3.6.29: Check the formula for S T in Example 3.60, by performing the sugges...
 3.3.2.29: Prove that the product of two upper triangular n n matrices is uppe...
 3.3.7.29: determine whether the given consumption matrix is productive.
 3.3.3.30: Is there an elementary matrix E such that EA D?Why or why not?
 3.3.4.30: Prove that every unit lower triangular matrix isinvertible and that...
 3.3.5.30: find a basis for the span of the given vectors
 3.3.1.30: assume that the product AB makessense. Prove that if the columns of...
 3.3.6.30: verify Theorem 3.32 by finding the matrix of S T (a) by direct subs...
 3.3.2.30: Prove Theorem 3.4(a)(c).
 3.3.7.30: determine whether the given consumption matrix is productive.
 3.3.3.31: find the inverse of the given elementary matrix
 3.3.4.31: A in Exercise 1
 3.3.5.31: find bases for the spans of the vectors in the given exercises from...
 3.3.1.31: compute AB by block multiplication,using the indicated partitioning.
 3.3.6.31: verify Theorem 3.32 by finding the matrix of S T (a) by direct subs...
 3.3.2.31: Prove Theorem 3.4(e).
 3.3.7.31: a consumption matrix C and a demand vector d are given. In each cas...
 3.3.3.32: find the inverse of the given elementary matrix
 3.3.4.32: A in Exercise 4
 3.3.5.32: find bases for the spans of the vectors in the given exercises from...
 3.3.1.32: compute AB by block multiplication,using the indicated partitioning.
 3.3.6.32: verify Theorem 3.32 by finding the matrix of S T (a) by direct subs...
 3.3.2.32: Using induction, prove that for all n
 3.3.7.32: a consumption matrix C and a demand vector d are given. In each cas...
 3.3.3.33: find the inverse of the given elementary matrix
 3.3.4.33: If A is symmetric and invertible and has an LDUfactorization, show ...
 3.3.5.33: Prove that if R is a matrix in echelon form, then a basis for row(R...
 3.3.1.33: compute AB by block multiplication,using the indicated partitioning.
 3.3.6.33: verify Theorem 3.32 by finding the matrix of S T (a) by direct subs...
 3.3.2.33: Using induction, prove that for all n 1, (A1 A2 An) T
 3.3.7.33: a consumption matrix C and a demand vector d are given. In each cas...
 3.3.3.34: find the inverse of the given elementary matrix
 3.3.4.34: If A is symmetric and invertible and A LDLT (with Lunit lower trian...
 3.3.5.34: Prove that if the columns of A are linearly independent,then they m...
 3.3.1.34: compute AB by block multiplication,using the indicated partitioning.
 3.3.6.34: verify Theorem 3.32 by finding the matrix of S T (a) by direct subs...
 3.3.2.34: Prove Theorem 3.5(b).
 3.3.7.34: a consumption matrix C and a demand vector d are given. In each cas...
 3.3.3.35: find the inverse of the given elementary matrix
 3.3.5.35: give the rank and the nullity of the matrices in the given exercise...
 3.3.1.35: Let(a) Compute(b) What is A2001? Why?
 3.3.6.35: verify Theorem 3.32 by finding the matrix of S T (a) by direct subs...
 3.3.2.35: (a) Prove that if A and B are symmetric n n matrices,then so is A B...
 3.3.7.35: Let A be an matrix, . Suppose that for some x in , . Prove that . x...
 3.3.3.36: find the inverse of the given elementary matrix
 3.3.5.36: give the rank and the nullity of the matrices in the given exercise...
 3.3.1.36: Let Find, with justification,B2011.
 3.3.6.36: find the standard matrix of the compositetransformation from 2 to 2...
 3.3.2.36: . (a) Give an example to show that if A and B aresymmetric n n matr...
 3.3.7.36: Let A, B, C, and D be matrices and x and y vectors in . Prove the f...
 3.3.3.37: find the inverse of the given elementary matrix
 3.3.5.37: give the rank and the nullity of the matrices in the given exercise...
 3.3.1.37: Let Find a formula for An (n 1) and verify your formula using mathe...
 3.3.6.37: find the standard matrix of the compositetransformation from 2 to 2...
 3.3.2.37: Which of the following matrices are skewsymmetric? 12105250 0 3 13...
 3.3.7.37: A population with three age classes has a Leslie matrix If the init...
 3.3.3.38: find the inverse of the given elementary matrix
 3.3.5.38: give the rank and the nullity of the matrices in the given exercise...
 3.3.1.38: Let (a) Show that (b) Prove, by mathematical induction, that An for...
 3.3.6.38: find the standard matrix of the compositetransformation from 2 to 2...
 3.3.2.38: Give a componentwise definition of a skewsymmetricmatrix.
 3.3.7.38: A population with four age classes has a Leslie matrix If the initi...
 3.3.3.39: find a sequence of elementary matricesE1, E2,..., Ek such that Ek E...
 3.3.5.39: If A is a 3 5 matrix, explain why the columns of Amust be linearly ...
 3.3.1.39: In each of the following, find the 6 6 matrix A [aij] that satisfie...
 3.3.6.39: find the standard matrix of the compositetransformation from 2 to 2...
 3.3.2.39: Prove that the main diagonal of a skewsymmetricmatrix must consist...
 3.3.7.39: A certain species with two age classes of 1 years duration has a su...
 3.3.3.40: find a sequence of elementary matricesE1, E2,..., Ek such that Ek E...
 3.3.5.40: If A is a 4 2 matrix, explain why the rows of A mustbe linearly dep...
 3.3.1.40: In each of the following, find the 6 6 matrix A [aij] that satisfie...
 3.3.6.40: use matrices to prove the given statements about transformations fr...
 3.3.2.40: Prove that if A and B are skewsymmetric n nmatrices, then so is A B.
 3.3.7.40: Suppose the Leslie matrix for the VW beetle is L Starting with an a...
 3.3.3.41: . Prove Theorem 3.13 for the case of AB I
 3.3.5.41: If A is a 3 5 matrix, what are the possible values of nullity(A)? 4
 3.3.1.41: Prove Theorem 3.1(a). aij b 1 if 6 i j 8 0 otherwise aij b 1 if 0 i...
 3.3.6.41: use matrices to prove the given statements about transformations fr...
 3.3.2.41: If A and B are skewsymmetric 2 2 matrices, underwhat conditions is...
 3.3.7.41: Suppose the Leslie matrix for the VW beetle is Investigate the effe...
 3.3.3.42: (a) Prove that if A is invertible and AB O, then B O. (b) Give a co...
 3.3.5.42: If A is a 4 2 matrix, what are the possible values of nullity(A)?
 3.3.6.42: use matrices to prove the given statements about transformations fr...
 3.3.2.42: Prove that if A is an n n matrix, then A AT isskewsymmetric.4
 3.3.7.42: Woodland caribou are found primarily in the western provinces of Ca...
 3.3.7.43: determine the adjacency matrix of the given graph.
 3.3.3.43: (a) Prove that if A is invertible and BA CA, then B C. (b) Give a c...
 3.3.5.43: find all possible values of rank(A) as a varies. A 1 2 a2 4a 2a 2 1 3
 3.3.6.43: use matrices to prove the given statements about transformations fr...
 3.3.2.43: (a) Prove that any square matrix A can be written asthe sum of a sy...
 3.3.7.44: determine the adjacency matrix of the given graph.
 3.3.3.44: A square matrix A is called idempotent if A2 A. (The word idempoten...
 3.3.5.44: find all possible values of rank(A) as a varies. a 2 13 3 22 1 aA 1 2
 3.3.6.44: Let T be a linear transformation from 2 to 2 (or from 3 to 3 ). Pro...
 3.3.2.44: If A and B are n n matrices, prove the followingproperties of the t...
 3.3.7.45: determine the adjacency matrix of the given graph.
 3.3.3.45: Show that if A is a square matrix that satisfies the equation A2 2A...
 3.3.5.45: considering the matrix with the given vectors as its columns. . Do ...
 3.3.6.45: Let T be a linear transformation from 2 to 2 (orfrom 3 to 3). Prove...
 3.3.2.45: Prove that if A and B are n n matrices, then tr(AB) tr(BA).
 3.3.7.46: determine the adjacency matrix of the given graph.
 3.3.3.46: Prove that if a symmetric matrix is invertible, then its inverse is...
 3.3.5.46: considering the matrix with the given vectors as its columns. Do fo...
 3.3.6.46: verify Theorem 3.32 by finding the matrix of S T (a) by direct subs...
 3.3.2.46: If A is any matrix, to what is tr(AAT) equal?
 3.3.7.47: draw a graph that has the given adjacency matrix.
 3.3.3.47: Prove that if A and B are square matrices and AB is invertible, the...
 3.3.5.47: considering the matrix with the given vectors as its columns. Do fo...
 3.3.6.47: verify Theorem 3.32 by finding the matrix of S T (a) by direct subs...
 3.3.2.47: Show that there are no 2 2 matrices A and B such that AB BA I2. t
 3.3.7.48: draw a graph that has the given adjacency matrix.
 3.3.3.48: use the GaussJordan method to find the inverse of the given matrix...
 3.3.5.48: considering the matrix with the given vectors as its columns. Do fo...
 3.3.6.48: verify Theorem 3.32 by finding the matrix of S T (a) by direct subs...
 3.3.7.49: draw a graph that has the given adjacency matrix.
 3.3.3.49: use the GaussJordan method to find the inverse of the given matrix...
 3.3.5.49: Do form a basis for ?
 3.3.6.49: verify Theorem 3.32 by finding the matrix of S T (a) by direct subs...
 3.3.7.50: draw a graph that has the given adjacency matrix.
 3.3.3.50: use the GaussJordan method to find the inverse of the given matrix...
 3.3.5.50: Do form a basis for ?
 3.3.6.50: verify Theorem 3.32 by finding the matrix of S T (a) by direct subs...
 3.3.7.51: determine the adjacency matrix of the given digraph.
 3.3.3.51: use the GaussJordan method to find the inverse of the given matrix...
 3.3.5.51: show that w is in span( ) and find the coordinate vector . B 120, 1...
 3.3.6.51: verify Theorem 3.32 by finding the matrix of S T (a) by direct subs...
 3.3.7.52: determine the adjacency matrix of the given digraph.
 3.3.3.52: use the GaussJordan method to find the inverse of the given matrix...
 3.3.5.52: show that w is in span( ) and find the coordinate vector . B 314, 5...
 3.3.6.52: Prove that P/ (cv) cP/ (v) for any scalar c [Example 3.59(b)].
 3.3.7.53: determine the adjacency matrix of the given digraph.
 3.3.3.53: use the GaussJordan method to find the inverse of the given matrix...
 3.3.5.53: compute the rank and nullity of the given matrices over the indicat...
 3.3.6.53: Prove that T : n S m is a linear transformation if and only if f
 3.3.7.54: determine the adjacency matrix of the given digraph.
 3.3.3.54: use the GaussJordan method to find the inverse of the given matrix...
 3.3.5.54: compute the rank and nullity of the given matrices over the indicat...
 3.3.6.54: Prove that (as noted at the beginning of this section) the range of...
 3.3.7.55: draw a digraph that has the given adjacency matrix
 3.3.3.55: use the GaussJordan method to find the inverse of the given matrix...
 3.3.5.55: compute the rank and nullity of the given matrices over the indicat...
 3.3.6.55: If A is an invertible 2 2 matrix, what does the Fundamental Theorem...
 3.3.7.56: draw a digraph that has the given adjacency matrix
 3.3.3.56: use the GaussJordan method to find the inverse of the given matrix...
 3.3.5.56: compute the rank and nullity of the given matrices over the indicat...
 3.3.7.57: draw a digraph that has the given adjacency matrix
 3.3.3.57: use the GaussJordan method to find the inverse of the given matrix...
 3.3.5.57: If A is , prove that every vector in null(A) is orthogonal to every...
 3.3.7.58: draw a digraph that has the given adjacency matrix
 3.3.3.58: use the GaussJordan method to find the inverse of the given matrix...
 3.3.5.58: If A and B are matrices of rank n, prove that AB has rank n.
 3.3.7.59: use powers of adjacency matrices to determine the number of paths o...
 3.3.3.59: use the GaussJordan method to find the inverse of the given matrix...
 3.3.5.59: (a) Prove that rank(AB) rank(B). [Hint: Review Exercise 29 in Secti...
 3.3.7.60: use powers of adjacency matrices to determine the number of paths o...
 3.3.3.60: use the GaussJordan method to find the inverse of the given matrix...
 3.3.5.60: (a) Prove that rank(AB) rank(A). [Hint: Review Exercise 30 in Secti...
 3.3.7.61: use powers of adjacency matrices to determine the number of paths o...
 3.3.3.61: use the GaussJordan method to find the inverse of the given matrix...
 3.3.5.61: (a) Prove that if U is invertible, then rank(UA) rank(A). [Hint: A ...
 3.3.7.62: use powers of adjacency matrices to determine the number of paths o...
 3.3.3.62: use the GaussJordan method to find the inverse of the given matrix...
 3.3.5.62: Prove that an matrix A has rank 1 if and only if A can be written a...
 3.3.7.63: use powers of adjacency matrices to determine the number of paths o...
 3.3.3.63: use the GaussJordan method to find the inverse of the given matrix...
 3.3.5.63: If an m n matrix A has rank r, prove that A can bewritten as the su...
 3.3.7.64: use powers of adjacency matrices to determine the number of paths o...
 3.3.3.64: c A B O Dd 1 c A1 A1 BD1 O D1 d 150
 3.3.5.64: Prove that, for m n matrices A and B, rank (A B) rank(A) rank(B).
 3.3.7.65: use powers of adjacency matrices to determine the number of paths o...
 3.3.3.65: c O B C I d 1 c 1BC2 1 1BC2 1 B C1BC2 1 I C1BC2 1 B d In Exer
 3.3.5.65: Let A be an n n matrix such that A2 O. Prove thatrank(A) n2. [Hint:...
 3.3.7.66: use powers of adjacency matrices to determine the number of paths o...
 3.3.3.66: c I B C I d 1 c 1I BC2 1 1I BC2 1 B C1I BC2 1 I C1I BC2 1 B d c O B C
 3.3.5.66: Let A be a skewsymmetric n n matrix.(See page 168)(a) Prove that x...
 3.3.7.67: Let A be the adjacency matrix of a graph G. (a) If row i of A is al...
 3.3.3.67: c O B C Dd 1 c c 1BD1C21 1BD1C21BD1D1C1BD1C21 D1 D1C1BD1C21BD1dcO B...
 3.3.7.68: Let A be the adjacency matrix of a digraph D. (a) If row i of A2 is...
 3.3.3.68: where P (A BD1C)1,Q PBD1, R D1CP, and S D1 D1CPBD1cP QR S c d, A BC
 3.3.7.69: Figure 3.29 is the digraph of a tournament with six players, P1 to ...
 3.3.3.69: partition the given matrix so that youcan apply one of the formulas...
 3.3.7.70: Figure 3.30 is a digraph representing a food web in a small ecosyst...
 3.3.3.70: partition the given matrix so that youcan apply one of the formulas...
 3.3.7.71: Five people are all connected by email. Whenever one of them hears...
 3.3.3.71: partition the given matrix so that youcan apply one of the formulas...
 3.3.7.72: Let A be the adjacency matrix of a graph G. (a) By induction, prove...
 3.3.7.73: If A is the adjacency matrix of a digraph G, what does the (i, j) e...
 3.3.7.74: determine whether a graph with the given adjacency matrix is bipart...
 3.3.7.75: determine whether a graph with the given adjacency matrix is bipart...
 3.3.7.76: determine whether a graph with the given adjacency matrix is bipart...
 3.3.7.77: determine whether a graph with the given adjacency matrix is bipart...
 3.3.7.78: a) Prove that a graph is bipartite if and only if its vertices can ...
 3.3.7.79: Suppose we encode the four vectors in by repeating the vector twice...
 3.3.7.80: Suppose we encode the binary digits 0 and 1 by repeating each digit...
 3.3.7.81: What is the result of encoding the messages in Exercises 8183 using...
 3.3.7.82: What is the result of encoding the messages in Exercises 8183 using...
 3.3.7.83: What is the result of encoding the messages in Exercises 8183 using...
 3.3.7.84: c 301001014 T
 3.3.7.85: c 311001104 T
 3.3.7.86: c 300111104 T
 3.3.7.87: The parity check code in Example 1.37 is a code (a) Find a standard...
 3.3.7.88: Define a (b) Find the associated standard parity check matrixfor th...
 3.3.7.89: Define a code using the standard generator matrix a) List all eight...
 3.3.7.90: Show that the code in Example 3.70 is a (3, 1) Hamming code.
 3.3.7.91: Construct standard parity check and generator matrices for a (15, 1...
 3.3.7.92: In Theorem 3.37, prove that if B A, then PGx 0 for every x in k2.
 3.3.7.93: . In Theorem 3.37, prove that if pi pj , then we cannot determine w...
Solutions for Chapter 3: Matrices
Full solutions for Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign)  3rd Edition
ISBN: 9780538735452
Solutions for Chapter 3: Matrices
Get Full SolutionsSince 407 problems in chapter 3: Matrices have been answered, more than 11902 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign), edition: 3. Chapter 3: Matrices includes 407 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) was written by and is associated to the ISBN: 9780538735452.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.